The LCM of two consecutive numbers is greater than 200 and is a multiple of 7. Name the numbers. How do you figure this out?

The first multiple of 7 greater than 200 is 203

I know that √203 = appr 14.25

so if I take 14 x 15 to get 210

I get a multiple of 7 and a number > 200

I have given you the smallest such pair

of course there would be an infinite number of such pairs.
e.g.
21x22
28x29
are only 2 examples of consecutive numbers whose LCM is greater than 200 and that LCM is a multiple of 7

Well, if we're talking about two consecutive numbers, let's call them X and X+1. The least common multiple (LCM) of X and X+1 is the smallest multiple that both X and X+1 share.

We know that the LCM is greater than 200 and is a multiple of 7. So, we can start by finding the LCM of two numbers greater than 200 that are multiples of 7.

If we take 210, for example, the next multiple of 7 is 217. Now, we check if 210 is divisible by (210 / 7) = 30, which it is. So, the LCM of 210 and 217 is 210.

Therefore, the numbers X and X+1 would be 210 and 211, respectively. Ta-da!

To find the numbers, we need to understand that the least common multiple (LCM) of two consecutive numbers will be the larger number itself. Since the LCM is greater than 200 and is a multiple of 7, we are looking for a number that satisfies these conditions.

Let's start by considering multiples of 7 greater than 200 until we find a consecutive pair:

1. 7: Not a consecutive pair.
2. 14: Not a consecutive pair.
3. 21: Not a consecutive pair.
4. 28: Not a consecutive pair.
5. 35: Not a consecutive pair.
6. 42: Not a consecutive pair.
7. 49: Not a consecutive pair.
8. 56: Not a consecutive pair.
9. 63: Not a consecutive pair.
10. 70: Not a consecutive pair.
11. 77: Not a consecutive pair.
12. 84: Not a consecutive pair.
13. 91: Not a consecutive pair.
14. 98: Not a consecutive pair.
15. 105: Not a consecutive pair.
16. 112: Not a consecutive pair.
17. 119: Not a consecutive pair.
18. 126: Not a consecutive pair.
19. 133: Not a consecutive pair.
20. 140: Not a consecutive pair.
21. 147: Not a consecutive pair.
22. 154: Not a consecutive pair.
23. 161: Not a consecutive pair.
24. 168: Not a consecutive pair.
25. 175: Not a consecutive pair.
26. 182: Not a consecutive pair.
27. 189: Not a consecutive pair.
28. 196: Not a consecutive pair.
29. 203: Not a multiple of 7.
30. 210: A consecutive pair.

Therefore, the numbers we are looking for are 210 and 211.

To find the two consecutive numbers, we need to determine the least common multiple (LCM) that satisfies the given conditions.

First, let's consider the conditions:

1) The LCM is greater than 200.
2) The LCM is a multiple of 7.

To find the LCM, we need to identify a common multiple of the two consecutive numbers. Given that the two numbers are consecutive, we can write them as n and n+1.

Next, let's determine the multiples of 7 that are greater than 200 and work our way up:

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ...

Based on the multiples of 7, we need to find a number that satisfies the condition of being greater than 200. From the list, we see that the first multiple that meets this criterion is 210.

Now, we can form an equation using the LCM formula:

LCM(n, n+1) = First multiple of 7 greater than 200

LCM(n, n+1) = 210

To solve this equation, we need to find two consecutive numbers whose LCM is 210. We can use factors or prime factorization to determine these numbers.

Prime factorization of 210: 2 x 3 x 5 x 7

Now, we can write down the factors of n and n+1 and find the two numbers:

n = 2 x 3 x 5 x 7
n+1 = 1 x 210 (since 210 has already been factored completely)

Simplifying:
n = 210
n+1 = 211

Therefore, the two consecutive numbers that satisfy the given conditions are 210 and 211.